For normalized sums \(Z_n=S_n/\sqrt{n}\) , Edgeworth expansions with non-integer moment conditions and non-uniform bounds have been investigated for a long time in the CLT and LLT (see e.g. [1]}, [2]}). In case of non-integer moments, a CLT for \(Z_n\)  has been derived in [3]} using fractional calculus. These stronger results lay the foundation for limit theorems in more powerful metrics such as expansions in the entropic CLT [4]} or related results for other information-theoretic distances [5]}, [6]}, [7]}, [8]}.
